DENVER—Amid national outbreaks of measles, whooping cough and other preventable diseases, Colorado officials might make it harder for parents to exempt children from vaccinations for school and day care.

Colorado is one of 48 states that allow such exemptions for reasons of personal belief or religion—often requiring little more than a parental signature on a form. In the 2012-2013 school year, about 4.3%, or 2,900 children, were...